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Hardcover ISBN:  9781470447809 
Product Code:  SURV/236 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470450618 
Product Code:  SURV/236.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470447809 
eBook ISBN:  9781470450618 
Product Code:  SURV/236.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 236; 2018; 427 ppMSC: Primary 37; 11; 28; 47
Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields.
Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results.
The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.
ReadershipGraduate students and researchers interested in ergodic theory and its connections to combinatorics and number theory.

Table of Contents

Chapters

Introduction

Part 1. Basics

Background material

Dynamical background

Rotations

Group extensions

Part 2. Cubes

Cubes in an algebraic setting

Dynamical cubes

Cubes in ergodic theory

The structure factors

Part 3. Nilmanifolds and nilsystems

Nilmanifolds

Nilsystems

Cubic structures in nilmanifolds

Factors of nilsystems

Polynomials in nilmanifolds and nilsystems

Arithmetic progressions in nilsystems

Part 4. Structure theorems

The ergodic structure theorem

Other structure theorems

Relations between consecutive factors

The structure theorem in a particular case

The structure theorem in the general case

Part 5. Applications

The method of characteristic factors

Uniformity seminorms on $\ell ^\infty $ and pointwise convergence of cubic averages

Multiple correlations, good weights, and antiuniformity

Inverse results for uniformity seminorms and applications

The comparison method


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Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields.
Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results.
The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.
Graduate students and researchers interested in ergodic theory and its connections to combinatorics and number theory.

Chapters

Introduction

Part 1. Basics

Background material

Dynamical background

Rotations

Group extensions

Part 2. Cubes

Cubes in an algebraic setting

Dynamical cubes

Cubes in ergodic theory

The structure factors

Part 3. Nilmanifolds and nilsystems

Nilmanifolds

Nilsystems

Cubic structures in nilmanifolds

Factors of nilsystems

Polynomials in nilmanifolds and nilsystems

Arithmetic progressions in nilsystems

Part 4. Structure theorems

The ergodic structure theorem

Other structure theorems

Relations between consecutive factors

The structure theorem in a particular case

The structure theorem in the general case

Part 5. Applications

The method of characteristic factors

Uniformity seminorms on $\ell ^\infty $ and pointwise convergence of cubic averages

Multiple correlations, good weights, and antiuniformity

Inverse results for uniformity seminorms and applications

The comparison method